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Pajara extensions: Difference between revisions

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{{Breadcrumb|Pajara}}
{{Breadcrumb|Pajara}}
The [[11-limit]] [[pajara]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:
* '''Tridecimal pajara''' ({{nowrap| 12 & 22 }}) – tempering out 50/49, 64/63, 65/63, 99/98;
* '''Pajarina''' ({{nowrap| 12f & 22 }}) – tempering out 50/49, 64/63, 78/77, 99/98;
* '''Pajarita''' ({{nowrap| 12f & 22f }}) – tempering out 40/39, 50/49, 64/63, 66/65;
In addition, we also consider the 13-limit extensions of important alternative 11-limit mappings. These are:
* '''Tridecimal pajarous''' ({{nowrap| 10 & 22 }}) – tempering out 50/49, 55/54, 64/63, 65/63;
* '''Pajaro''' ({{nowrap| 10 & 22f }}) – tempering out 40/39, 50/49, 55/54, 64/63.
* '''Pajaric''' ({{nowrap| 10 & 12f }}) – tempering out 40/39, 45/44, 50/49, 56/55.


== Tuning spectra ==
== Tuning spectra ==
=== Pajara ===
=== Pajara ===
 
{| class="wikitable center-all left-4"
[[Gencom|Gencom]]: [7/5 3/2; 50/49 64/63 65/63 99/98]
 
Gencom mapping: [{{map|2 2 7 8 14 5}}, {{map|0 1 -2 -2 -6 2}}]
 
{| class="wikitable center-all"
|-
|-
! | ET<br>generator
! Edo<br>generator
! | [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! | fifth<br>(¢)
! Generator (¢)
! | comments
! Comments
|-
|-
| |
|  
| | 4/3
| 4/3
| | 701.955
| 701.955
| |
|  
|-
|-
| |
|  
| | 14/11
| 14/11
| | 704.377
| 704.377
| |
|  
|-
|-
| |
|  
| | 10/9
| 10/9
| | 704.399
| 704.399
| |
|  
|-
|-
| |
|  
| | 6/5
| 6/5
| | 705.214
| 705.214
| | 5 limit minimax
| 5 limit minimax
|-
|-
| |
|  
| | 3125/2592
| 3125/2592
| | 705.447
| 705.447
| | 5 limit least squares
| 5 limit least squares
|-
|-
| | 20\34
| 20\34
| |
|  
| | 705.882
| 705.882
| |
|  
|-
|-
| |
|  
| | 11/9
| 11/9
| | 706.574
| 706.574
| |
|  
|-
|-
| |
|  
| | 5/4
| 5/4
| | 706.843
| 706.843
| |
|  
|-
|-
| |
|  
| | {{vector|0 -50 5 5 29}}
| {{vector|0 -50 5 5 29}}
| | 707.106
| 707.106
| | 11 limit least squares
| 11 limit least squares
|-
|-
| | 33\56
| 33\56
| |
|  
| | 707.143
| 707.143
| |
|  
|-
|-
| |
|  
| | 12/11
| 12/11
| | 707.234
| 707.234
| |
|  
|-
|-
| |
|  
| | 35/27
| 35/27
| | 707.246
| 707.246
| | 9 limit least squares
| 9 limit least squares
|-
|-
| |
|  
| | 15/11
| 15/11
| | 707.390
| 707.390
| |
|  
|-
|-
| |
|  
| | 11/8
| 11/8
| | 708.114
| 708.114
| |
|  
|-
|-
| |
|  
| | 11/10
| 11/10
| | 708.749
| 708.749
| | 11 limit minimax
| 11 limit minimax
|-
|-
| |
|  
| | 9/7
| 9/7
| | 708.771
| 708.771
| | 9 limit minimax
| 9 limit minimax
|-
|-
| |
|  
| | 71663616/52521875
| 71663616/52521875
| | 708.814
| 708.814
| | 7 limit least squares
| 7 limit least squares
|-
|-
| |
|  
| | {{vector|0 -49 9 9 37 -19}}
| {{vector|0 -49 9 9 37 -19}}
| | 708.901
| 708.901
| | 13 limit least squares
| 13 limit least squares
|-
|-
| |
|  
| | {{vector|0 -51 7 10 42 -22}}
| {{vector|0 -51 7 10 42 -22}}
| | 709.038
| 709.038
| | 15 limit least squares
| 15 limit least squares
|-
|-
| | 13\22
| 13\22
| |
|  
| | 709.091
| 709.091
| |
|  
|-
|-
| |
|  
| | 7/6
| 7/6
| | 711.043
| 711.043
| | 7 limit minimax
| 7 limit minimax
|-
|-
| | 32\54
| 32\54
| |
|  
| | 711.111
| 711.111
| |
|  
|-
|-
| |
|  
| | 13/11
| 13/11
| | 711.151
| 711.151
| | 13 and 15 limit minimax
| 13 and 15 limit minimax
|-
|-
| |
|  
| | 16/15
| 16/15
| | 711.731
| 711.731
| |
|  
|-
|-
| | 19\32
| 19\32
| |
|  
| | 712.500
| 712.500
| |
|  
|-
|-
| |
|  
| | 13/10
| 13/10
| | 713.553
| 713.553
| |
|  
|-
|-
| |
|  
| | 8/7
| 8/7
| | 715.587
| 715.587
| |
|  
|-
|-
| |
|  
| | 15/13
| 15/13
| | 717.420
| 717.420
| |
|  
|-
|-
| |
|  
| | 14/13
| 14/13
| | 717.925
| 717.925
| |
|  
|-
|-
| |
|  
| | 15/14
| 15/14
| | 719.443
| 719.443
| |
|  
|-
|-
| |
|  
| | 16/13
| 16/13
| | 720.264
| 720.264
| |
|  
|-
|-
| |
|  
| | 13/12
| 13/12
| | 738.573
| 738.573
| |
|  
|}
|}


=== Pajarous ===
=== Pajarous ===
 
{| class="wikitable center-all left-4"
[[Gencom|Gencom]]: [7/5 3/2; 50/49 55/54 64/63 65/63]
 
Gencom mapping: [{{map|2 2 7 8 1 5}}, {{map|0 1 -2 -2 5 2}}]
 
{| class="wikitable center-all"
|-
|-
! | ET<br>generator
! Edo<br>generator
! | eigenmonzo<br>(unchanged interval)
! Eigenmonzo<br>(unchanged interval)
! | fifth<br>(¢)
! Generator (¢)
! | comments
! Comments
|-
|-
| |
|  
| | 4/3
| 4/3
| | 701.955
| 701.955
| |
|  
|-
|-
| |
|  
| | 13/11
| 13/11
| | 703.597
| 703.597
| |
|  
|-
|-
| |
|  
| | 10/9
| 10/9
| | 704.399
| 704.399
| |
|  
|-
|-
| |
|  
| | 6/5
| 6/5
| | 705.214
| 705.214
| | 5 limit minimax
| 5 limit minimax
|-
|-
| |
|  
| | 3125/2592
| 3125/2592
| | 705.447
| 705.447
| | 5 limit least squares
| 5 limit least squares
|-
|-
| | 20\34
| 20\34
| |
|  
| | 705.882
| 705.882
| |
|  
|-
|-
| |
|  
| | 5/4
| 5/4
| | 706.843
| 706.843
| |
|  
|-
|-
| | 33\56
| 33\56
| |
|  
| | 707.143
| 707.143
| |
|  
|-
|-
| |
|  
| | 35/27
| 35/27
| | 707.246
| 707.246
| | 9 limit least squares
| 9 limit least squares
|-
|-
| |
|  
| | 9/7
| 9/7
| | 708.771
| 708.771
| | 9, 13 and 15 limit minimax
| 9, 13 and 15 limit minimax
|-
|-
| |
|  
| | 71663616/52521875
| 71663616/52521875
| | 708.814
| 708.814
| | 7 limit least squares
| 7 limit least squares
|-
|-
| | 13\22
| 13\22
| |
|  
| | 709.091
| 709.091
| |
|  
|-
|-
| |
|  
| | 11/10
| 11/10
| | 709.286
| 709.286
| | 11 limit minimax
| 11 limit minimax
|-
|-
| |
|  
| | {{vector|0 17 -16 -16 26}}
| {{vector|0 17 -16 -16 26}}
| | 709.883
| 709.883
| | 11 limit least squares
| 11 limit least squares
|-
|-
| |
|  
| | 11/8
| 11/8
| | 710.264
| 710.264
| |
|  
|-
|-
| |
|  
| | 15/11
| 15/11
| | 710.508
| 710.508
| |
|  
|-
|-
| |
|  
| | {{vector|0 16 -20 -20 29 8}}
| {{vector|0 16 -20 -20 29 8}}
| | 710.665
| 710.665
| | 13 limit least squares
| 13 limit least squares
|-
|-
| |
|  
| | {{vector|0 7 -29 -21 35 11}}
| {{vector|0 7 -29 -21 35 11}}
| | 710.879
| 710.879
| | 15 limit least squares
| 15 limit least squares
|-
|-
| |
|  
| | 7/6
| 7/6
| | 711.043
| 711.043
| | 7 limit minimax
| 7 limit minimax
|-
|-
| | 32\54
| 32\54
| |
|  
| | 711.111
| 711.111
| |
|  
|-
|-
| |
|  
| | 16/15
| 16/15
| | 711.731
| 711.731
| |
|  
|-
|-
| |
|  
| | 14/11
| 14/11
| | 711.785
| 711.785
| |
|  
|-
|-
| |
|  
| | 12/11
| 12/11
| | 712.341
| 712.341
| |
|  
|-
|-
| | 19\32
| 19\32
| |
|  
| | 712.500
| 712.500
| |
|  
|-
|-
| |
|  
| | 13/10
| 13/10
| | 713.553
| 713.553
| |
|  
|-
|-
| |
|  
| | 8/7
| 8/7
| | 715.587
| 715.587
| |
|  
|-
|-
| |
|  
| | 11/9
| 11/9
| | 715.803
| 715.803
| |
|  
|-
|-
| |
|  
| | 15/13
| 15/13
| | 717.420
| 717.420
| |
|  
|-
|-
| |
|  
| | 14/13
| 14/13
| | 717.925
| 717.925
| |
|  
|-
|-
| |
|  
| | 15/14
| 15/14
| | 719.443
| 719.443
| |
|  
|-
|-
| |
|  
| | 16/13
| 16/13
| | 720.264
| 720.264
| |
|  
|-
|-
| |
|  
| | 13/12
| 13/12
| | 738.573
| 738.573
| |
|  
|}
|}


=== Pajaro ===
=== Pajaro ===
 
{| class="wikitable center-all left-4"
[[Gencom|Gencom]]: [7/5 3/2; 40/39 50/49 55/54 64/63]
 
Gencom mapping: [{{map|2 2 7 8 1 11}}, {{map|0 1 -2 -2 5 -3}}]
 
{| class="wikitable center-all"
|-
|-
! | ET<br>generator
! Edo<br>generator
! | eigenmonzo<br>(unchanged interval)
! Eigenmonzo<br>(unchanged interval)
! | fifth<br>(¢)
! Generator (¢)
! | comments
! Comments
|-
|-
| |
|  
| | 4/3
| 4/3
| | 701.955
| 701.955
| |
|  
|-
|-
| |
|  
| | 10/9
| 10/9
| | 704.399
| 704.399
| |
|  
|-
|-
| |
|  
| | 6/5
| 6/5
| | 705.214
| 705.214
| | 5 limit minimax
| 5 limit minimax
|-
|-
| |
|  
| | 3125/2592
| 3125/2592
| | 705.447
| 705.447
| | 5 limit least squares
| 5 limit least squares
|-
|-
| | 20\34
| 20\34
| |
|  
| | 705.882
| 705.882
| |
|  
|-
|-
| |
|  
| | 5/4
| 5/4
| | 706.843
| 706.843
| |
|  
|-
|-
| | 33\56
| 33\56
| |
|  
| | 707.143
| 707.143
| |
|  
|-
|-
| |
|  
| | 35/27
| 35/27
| | 707.246
| 707.246
| | 9 limit least squares
| 9 limit least squares
|-
|-
| |
|  
| | 9/7
| 9/7
| | 708.771
| 708.771
| | 9 limit minimax
| 9 limit minimax
|-
|-
| |
|  
| | 71663616/52521875
| 71663616/52521875
| | 708.814
| 708.814
| | 7 limit least squares
| 7 limit least squares
|-
|-
| | 13\22
| 13\22
| |
|  
| | 709.091
| 709.091
| |
|  
|-
|-
| |
|  
| | 11/10
| 11/10
| | 709.286
| 709.286
| | 11 limit minimax
| 11 limit minimax
|-
|-
| |
|  
| | {{vector|0 17 -16 -16 26}}
| {{vector|0 17 -16 -16 26}}
| | 709.883
| 709.883
| | 11 limit least squares
| 11 limit least squares
|-
|-
| |
|  
| | 11/8
| 11/8
| | 710.264
| 710.264
| |
|  
|-
|-
| |
|  
| | 15/11
| 15/11
| | 710.508
| 710.508
| |
|  
|-
|-
| |
|  
| | 7/6
| 7/6
| | 711.043
| 711.043
| | 7 limit minimax
| 7 limit minimax
|-
|-
| | 32\54
| 32\54
| |
|  
| | 711.111
| 711.111
| |
|  
|-
|-
| |
|  
| | {{vector|0 31 -15 -15 34 -22}}
| {{vector|0 31 -15 -15 34 -22}}
| | 711.580
| 711.580
| | 13 limit least squares
| 13 limit least squares
|-
|-
| |
|  
| | {{vector|0 27 -19 -16 40 -24}}
| {{vector|0 27 -19 -16 40 -24}}
| | 711.631
| 711.631
| | 15 limit least squares
| 15 limit least squares
|-
|-
| |
|  
| | 16/15
| 16/15
| | 711.731
| 711.731
| |
|  
|-
|-
| |
|  
| | 14/11
| 14/11
| | 711.785
| 711.785
| |
|  
|-
|-
| |
|  
| | 12/11
| 12/11
| | 712.341
| 712.341
| |
|  
|-
|-
| | 19\32
| 19\32
| |
|  
| | 712.500
| 712.500
| |
|  
|-
|-
| |
|  
| | 18/13
| 18/13
| | 712.676
| 712.676
| | 13 and 15 limit minimax
| 13 and 15 limit minimax
|-
|-
| |
|  
| | 13/11
| 13/11
| | 713.849
| 713.849
| |
|  
|-
|-
| |
|  
| | 13/12
| 13/12
| | 715.357
| 715.357
| |
|  
|-
|-
| |
|  
| | 8/7
| 8/7
| | 715.587
| 715.587
| |
|  
|-
|-
| |
|  
| | 11/9
| 11/9
| | 715.803
| 715.803
| |
|  
|-
|-
| |
|  
| | 15/14
| 15/14
| | 719.443
| 719.443
| |
|  
|-
|-
| |
|  
| | 16/13
| 16/13
| | 719.824
| 719.824
| |
|  
|-
|-
| |
|  
| | 15/13
| 15/13
| | 723.871
| 723.871
| |
|  
|-
|-
| |
|  
| | 14/13
| 14/13
| | 728.298
| 728.298
| |
|  
|-
|-
| |
|  
| | 13/10
| 13/10
| | 745.786
| 745.786
| |
|  
|}
|}


=== Pajaric ===
=== Pajaric ===
 
{| class="wikitable center-all left-4"
[[Gencom|Gencom]]: [7/5 3/2; 40/39 45/44 50/49 56/55]
 
Gencom mapping: [{{map|2 2 7 8 7 11}}, {{map|0 1 -2 -2 0 -3}}]
 
{| class="wikitable center-all"
|-
|-
! | ET<br>generator
! Edo<br>generator
! | eigenmonzo<br>(unchanged interval)
! Eigenmonzo<br>(unchanged interval)
! | fifth<br>(¢)
! Generator (¢)
! | comments
! Comments
|-
|-
| |
|  
| | 15/11
| 15/11
| | 663.049
| 663.049
| |
|  
|-
|-
| |
|  
| | 11/10
| 11/10
| | 682.502
| 682.502
| |
|  
|-
|-
| |
|  
| | 14/11
| 14/11
| | 691.246
| 691.246
| |
|  
|-
|-
| |
|  
| | 4/3
| 4/3
| | 701.955
| 701.955
| |
|  
|-
|-
| |
|  
| | 13/11
| 13/11
| | 703.597
| 703.597
| |
|  
|-
|-
| |
|  
| | 10/9
| 10/9
| | 704.399
| 704.399
| |
|  
|-
|-
| |
|  
| | 6/5
| 6/5
| | 705.214
| 705.214
| | 5 limit minimax
| 5 limit minimax
|-
|-
| |
|  
| | 3125/2592
| 3125/2592
| | 705.447
| 705.447
| | 5 limit least squares
| 5 limit least squares
|-
|-
| | 20\34
| 20\34
| |
|  
| | 705.882
| 705.882
| |
|  
|-
|-
| |
|  
| | {{vector|0 -32 11 11 -1}}
| {{vector|0 -32 11 11 -1}}
| | 706.676
| 706.676
| | 11 limit least squares
| 11 limit least squares
|-
|-
| |
|  
| | 5/4
| 5/4
| | 706.843
| 706.843
| | 11, 13 and 15 limit minimax
| 11, 13 and 15 limit minimax
|-
|-
| | 33\56
| 33\56
| |
|  
| | 707.143
| 707.143
| |
|  
|-
|-
| |
|  
| | 35/27
| 35/27
| | 707.246
| 707.246
| | 9 limit least squares
| 9 limit least squares
|-
|-
| |
|  
| | 9/7
| 9/7
| | 708.771
| 708.771
| | 9 limit minimax
| 9 limit minimax
|-
|-
| |
|  
| | 71663616/52521875
| 71663616/52521875
| | 708.814
| 708.814
| | 7 limit least squares
| 7 limit least squares
|-
|-
| | 13\22
| 13\22
| |
|  
| | 709.091
| 709.091
| |
|  
|-
|-
| |
|  
| | {{vector|0 -46 10 10 -4 17}}
| {{vector|0 -46 10 10 -4 17}}
| | 709.889
| 709.889
| | 13 limit least squares
| 13 limit least squares
|-
|-
| |
|  
| | {{vector|0 -47 9 11 -5 19}}
| {{vector|0 -47 9 11 -5 19}}
| | 710.032
| 710.032
| | 15 limit least squares
| 15 limit least squares
|-
|-
| |
|  
| | 7/6
| 7/6
| | 711.043
| 711.043
| | 7 limit minimax
| 7 limit minimax
|-
|-
| | 32\54
| 32\54
| |
|  
| | 711.111
| 711.111
| |
|  
|-
|-
| |
|  
| | 16/15
| 16/15
| | 711.731
| 711.731
| |
|  
|-
|-
| | 19\32
| 19\32
| |
|  
| | 712.500
| 712.500
| |
|  
|-
|-
| |
|  
| | 18/13
| 18/13
| | 712.676
| 712.676
| |
|  
|-
|-
| |
|  
| | 13/12
| 13/12
| | 715.357
| 715.357
| |
|  
|-
|-
| |
|  
| | 8/7
| 8/7
| | 715.587
| 715.587
| |
|  
|-
|-
| |
|  
| | 15/14
| 15/14
| | 719.443
| 719.443
| |
|  
|-
|-
| |
|  
| | 16/13
| 16/13
| | 719.824
| 719.824
| |
|  
|-
|-
| |
|  
| | 15/13
| 15/13
| | 723.871
| 723.871
| |
|  
|-
|-
| |
|  
| | 11/9
| 11/9
| | 726.296
| 726.296
| |
|  
|-
|-
| |
|  
| | 14/13
| 14/13
| | 728.298
| 728.298
| |
|  
|-
|-
| |
|  
| | 13/10
| 13/10
| | 745.786
| 745.786
| |
|  
|-
|-
| |
|  
| | 12/11
| 12/11
| | 750.637
| 750.637
| |
|  
|}
|}


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